Matlab implements IEEE double precision floating point arithmetic. Each floating point number $x$ is of the form

$x=+-0.d_1{d}_{2}cdots{d}_{53}*2^e,-1023e1023.$

If $-1022e1023,x$ is assumed to be normalized, meaning that $d_1=1$; since $d_1$ is known to be $1,$ it need not be stored. To represent very small numbers, the following convention is adopted: If $e=-1023,$ then $d_2{d}_{3}cdots{d}_{53}$ is interpreted as $0\mathrm{.}0d_2{d}_{3}cdots{d}_{53}*2^{-1022}($no typo $-$ an exponent of $-1023$ is interpreted as an exponent of $-1022$ with $d_1=0).($Since $d_1$ is not stored, the bits make sense: $1$ bit for the sign, $52$ bits for $d_2,d_3,$dots,$d_{53},$ and $11$ bits for the exponent, for a total of $64$ bits, or $8$ bytes.$)$

$($a$)$ What are the largest and smallest positive floating points numbers in this system?

$($b$)$ What is the smallest floating point number strictly larger than $1$ and the largest floating point number strictly less than $1?$

$($c$)$ Write Matlab $($or Python or Julia or $...)$ code to verify the above conclusions. For example, if $x$ is the largest positive floating point number, your code should verify that $x$ is exactly representable and that any number larger than $x$ either overflows to Inf or is rounded down to $x.$ Similarly for the other three floating numbers described above.

Can you give me a detailed solution so that i can understand thank you